That current and voltage depend on t is implicit. If they did not there would be no need for differential equations since nothing would ever change. In cases like this you can expnad the short hand as follows:

i -> i(t) ; i(t) is a function of t, but we do not yet know the form of this function

The derivative of i(t) is just d/dt[ i(t) ], and since we don't know the form of i(t) we cannot know the form of di/dt. However, the differential equation puts a severe restriction of the form of i(t) and therefore di/dt. Once you see the form of a differential equation you can make an ansatz

Essentially, you've re-arranged things in the same manner as I did before making the change of variable substitution [z(t)=iL(t)-(E/R)] - which I did to allow me to more easily perform the integration of both sides.